Pith. sign in

REVIEW 5 major objections 4 minor 1 cited by

Scale Invariant Entanglement Negativity at the Many-Body Localization Transition

T0 review · 5 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read At the many-body localization transition, the logarithmic negativity between two blocks decays exponentially with the ratio of separation to block size, while mutual information decays as a power law, showing that critical eigenstates are…

desk verdict A clever two-length-scale probe of the MBL transition, but the scale-invariance claim rests on a self-selected critical point at a single system size. read the letter →

arxiv 1908.02761 v1 pith:S3I7IPE3 submitted 2019-08-07 cond-mat.dis-nn cond-mat.stat-mechquant-ph

Johnnie Gray , Abolfazl Bayat , Arijeet Pal , Sougato Bose This is my paper · ORCID
classification cond-mat.dis-nncond-mat.stat-mechquant-ph MSC 81P4082B2782B44
keywords many-bodylocalizationscaleinvariancelogarithmicnegativitymutualinformationentanglementtransitiondisorderedspinchaincriticaleigenstatestensornetworks
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that at the many-body localization (MBL) transition, the entanglement structure of critical eigenstates is scale invariant: the logarithmic negativity between two equal-sized blocks decays exponentially with the ratio of separation to block size, while the mutual information decays as a power law. The authors probe this by computing these quantities in a disordered spin chain for a range of block sizes and separations, and find that at disorder strength $h=3.25$ (the inferred transition point for $L=20$) the data for different block sizes collapses onto a single curve. The claim matters because scale invariance is a hallmark of a continuous phase transition, and identifying it in the entanglement structure constrains theories of the MBL transition and opens the door to tensor-network simulations of critical eigenstates.

What carries the argument

The central object is the pair of length scales, block size $l$ and separation $d$, combined into the normalized separation $\tilde{d}=d/l$. By averaging the normalized logarithmic negativity $\tilde{E}=E/l$ and mutual information $\tilde{I}=I/l$ over pairs of blocks with the same $\tilde{d}$, the authors test whether all dependence on $l$ disappears at the transition. The mechanism is the data collapse: at $h=3.25$ the curves for different $l$ converge onto a universal exponential for logarithmic negativity and a power law for mutual information, which is the signature of scale invariance.

What would settle it

Compute the same quantities for a larger chain, say $L=24$ or $L=28$, at several disorder strengths around $h=3.25$, and check whether the data collapse persists at a disorder strength that is independently identified as critical (for example, from level statistics or other standard probes); if no choice of $h$ yields collapse for all block sizes at larger $L$, the scale-invariant exponential and power-law ansatzes are falsified.

Watch

Extended reading notes

Core claim

The central discovery is that the bond logarithmic negativity and the mutual information between disjoint blocks of equal size $l$, expressed as functions of the normalized separation $\tilde{d}=d/l$, become independent of $l$ at the MBL transition point. Specifically, the logarithmic negativity decays exponentially as $\tilde{E}_{\tilde{d}}(l) = C_E e^{-\tilde{d}/\lambda_E}$ with fitted $C_E\approx 2$ and $\lambda_E\approx 0.45$, while the mutual information decays as $\tilde{I}_{\tilde{d}}(l) = C_I \tilde{d}^{-1/\alpha_I}$ with fitted $C_I\approx 0.3$ and $\alpha_I\approx 0.5$. This scale invariance is accompanied by a logarithmic growth of the average self-entanglement with block size at the transition, in contrast to linear growth in the ergodic phase and area-law (constant) behavior deep in the localized phase. The authors interpret this as evidence for a scale-invariant, multipartite entanglement structure in critical eigenstates.

Load-bearing premise

The transition point for $L=20$ is identified as $h=3.25$ from the same nearest-neighbor data collapse (Fig. 3) that is then used to demonstrate scale invariance in Fig. 4, so if the true critical point for that system size were different or shifts significantly with system size, the collapse could be coincidental.

Editorial extensions

If this is right

  • If correct, critical eigenstates near the MBL transition have a scale-invariant entanglement structure, meaning any theory of the transition must reproduce the exponential decay of negativity and the polynomial decay of mutual information with normalized separation.
  • The observed collapse provides a parameter-free way to identify the transition point in finite systems, independent of the usual finite-size scaling with total system size.
  • The logarithmic growth of self-entanglement at the transition suggests a connection to conformal field theory or infinite-randomness fixed points, and may support a multiscale entanglement renormalization ansatz (MERA) description of critical eigenstates, enabling tensor-network simulations beyond exact diagonalization.
  • The distinction between the decay of quantum correlations (exponential) and total correlations (power law) at the same point implies that the critical state is not captured by simple quantum-classical equivalences, constraining strong-disorder renormalization group approaches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step would be to check whether the scale-invariant collapse persists for larger system sizes than $L=20$; if it moves with $L$, the observed $h=3.25$ collapse might be a finite-size artifact rather than a true critical point.
  • The power-law decay of mutual information with exponent near $1/2$ could be related to a logarithmic growth of entanglement entropy when integrated over separations; testing the probability distribution of negativity instead of just the mean would further probe the multipartite structure.
  • The exponential decay length $\lambda_E\approx 0.45$ in units of $d/l$ might be interpretable as a critical correlation length in the ratio variable, which real-space renormalization group schemes should predict if they respect scale invariance.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper studies the scale-invariance of entanglement and correlation measures at the many-body localization (MBL) transition in a disordered Heisenberg chain. For L=20, exact diagonalization is used to compute the logarithmic negativity and mutual information between two disjoint blocks of equal size l separated by distance d. The authors report that at disorder strength h≈3.25, the normalized block measures collapse onto a single curve as a function of d/l, with logarithmic negativity decaying exponentially and mutual information decaying as a power law. They also report a logarithmic scaling of the self-entanglement near the transition and argue that these findings reveal a scale-invariant, multipartite entanglement structure in critical eigenstates, with implications for tensor-network descriptions of the MBL transition.

Significance. If correct, the result provides a concrete and falsifiable signature of scale invariance at the MBL transition and a clear distinction between the decay of logarithmic negativity and mutual information, which would constrain renormalization-group and tensor-network theories of the transition. The use of two controllable length scales (block size and separation) is a natural and powerful probe, and the visual collapse in Fig. 4 is striking. The paper also releases code (quimb) that enables independent verification of the numerics. However, the central claim rests on a single disorder strength at a single system size, with the critical field selected from the same data that is later used to claim collapse, and no quantitative collapse metrics or error estimates are provided. These issues temper the significance of the result as it stands.

major comments (5)
  1. [Results, Fig. 3 and text after Eq. (4)] The identification of h=3.25 as the transition point for L=20 is made by the observed collapse of the nearest-neighbor normalized quantities in Fig. 3, and the same value is then used in Fig. 4(b,e) to demonstrate the scale-invariant decay. This is a circular selection: the collapse in Fig. 4 is not an independent test but a consequence of choosing h to make the nearest-neighbor data collapse. The text states 'We infer that h∼3.25 corresponds to the transition point for this total system size of L=20, which matches previous studies [73],' but reference [73] is a software paper and does not provide an independent estimate of h_c. Since the Model section itself quotes the suspected transition range h~3.5-5, the manuscript needs an independent determination of h_c(L=20)—for example from level statistics or entanglement entropy scaling—before the collapse can be attributed to criticality. As written, the scale invariance claim is not tested but imposed.
  2. [Results, Fig. 4 and Eqs. (5)-(6)] The exponential and power-law fits to the collapsed data in Fig. 4 are reported without any uncertainties, goodness-of-fit measures, or comparison to alternative forms. The statement in the Fig. 4 caption that a stretched exponential 'was not found to be as natural as a power law' is not supported by any quantitative criterion. Furthermore, no error bars or collapse-quality metric (e.g., variance of the curves, a Q-test, or bootstrap analysis) are provided, so the reader cannot judge whether the residual l-dependence in Fig. 4(b,e) is statistically significant. A quantitative collapse measure is essential for the central claim of scale invariance.
  3. [Results, Fig. 4] The scale-invariance claim is demonstrated only at L=20. Although Fig. 5 examines the self entanglement for L=14,16,18,20 and shows a peak in the log-law coefficient, the bond quantities E~d(l) and I~d(l) are never analyzed as functions of L. Consequently, it is unknown whether the data collapse improves with system size or whether the fitted parameters (λE≈0.45, αI≈0.5) drift. Without a finite-size scaling analysis, the observed bundle of curves at one disorder strength and one system size cannot be distinguished from a finite-size crossing.
  4. [Results, Fig. 4] The accessible range of d/l depends strongly on the block size l because two disjoint blocks of size l in an open chain of length L satisfy d ≤ L-l. For L=20, l=10 admits only d/l=1, while l=1 admits d/l up to 19. Thus the large-d/l tail of the collapse in Fig. 4 is supported almost entirely by the smallest block sizes, and the exponential (Eq. 5) and power-law (Eq. 6) forms are not tested uniformly across all block sizes. The authors should either restrict the quoted functional forms to the common range of d/l for all l, or present the data separated by l to show the collapse holds in overlapping ranges.
  5. [Eqs. (5)-(6) and the paragraph after them] The text says that the coefficients CE, CI, λE, and αI 'might all be functions of L, l and h,' but then quotes single values (CE~2, λE~0.45, CI~0.3, αI~0.5). If these coefficients carry any l-dependence, the apparent collapse in Fig. 4 after normalizing by l would be a trivial scaling property rather than a universal signature. The authors must state explicitly that the fits in Eqs. (5)-(6) assume l-independent constants, and should report the fitting residuals as a function of l to demonstrate that the collapse is not an artifact of the normalization.
minor comments (4)
  1. [Eq. (3)] The partial transpose in the definition of logarithmic negativity is written as ρ_AB^{T_X} with 'subsystem X' undefined; it should specify TX = TA or TB.
  2. [Fig. 3 caption] The word 'Blocked' in the caption 'Average Nearest Neighbour Blocked Logarithmic Negativity' appears to be a typo; likely 'block' or 'block-entanglement' was intended.
  3. [Fig. 5 caption] The caption reports mean fitting uncertainties for the three coefficients but the main text does not refer to them; providing analogous uncertainties for the fits in Eqs. (5)-(6) would improve the manuscript.
  4. [Model section, paragraph 2] The sentence 'The transition point, h_c, between these two phases is suspected to lie between h∼3.5−5 [42,69]' is in tension with the later use of h=3.25; the discrepancy should be acknowledged and discussed in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; scale-invariance claim is an empirical collapse observation with h_c fixed by the same data family but not by construction.

full rationale

This paper is a numerical-observation study, not a derivation, and I find no step in which a claimed prediction reduces by construction to an input. The critical field h≈3.25 for L=20 is identified empirically from the nearest-neighbour collapse in Fig. 3 and then used in Fig. 4; although this is a self-referential way to locate the transition, it is not a circular reduction: the full d/l collapse in Fig. 4(b,e) is a much larger set of data than the d~=1 curves used to fix h_c, and the collapse is not guaranteed by the normalization. The exponential and power-law forms in Eqs. (5)-(6) are explicitly presented as ansatzes fitted to the data ('we suggest the following ansatzes', 'we perform least squares fitting'), not as predictions derived from the model. The self-citations [42] and [73] are used only to note consistency with previous estimates and numerical methods; the load-bearing evidence is the authors' own exact-diagonalization data. There is no imported uniqueness theorem, no ansatz smuggled in via citation, and no renaming of a known result. The main caveat—that h_c is not independently located by finite-size scaling—is a correctness/robustness concern, not a circularity.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The central scale invariance claim rests on one hand-picked critical field h_c=3.25, four fit coefficients describing the decay forms, and three fit coefficients for self-entanglement scaling. No new physical entities are postulated. The main unproved assumptions are the standard MBL model mapping, the representativeness of a single middle eigenstate, the accuracy of the TNSLQ large-block approximation, and the interpretive link between collapse and scale invariance.

free parameters (8)
  • h_c (critical disorder strength for L=20) = 3.25
    Identified as the point where nearest-neighbor normalized E and I curves collapse in Fig. 3, and stated to match previous studies. The scale invariance claim depends on this value.
  • lambda_E (decay length in Eq. 5) = ~0.45
    Least-squares fit of E_tilde_d = C_E exp(-d_tilde/lambda_E) to collapse data at h=3.25.
  • C_E (prefactor in Eq. 5) = ~2
    Fit coefficient for exponential decay of logarithmic negativity.
  • alpha_I (power-law exponent in Eq. 6) = ~0.5
    Least-squares fit of I_tilde_d = C_I d_tilde^{-1/alpha_I}, giving a power law with exponent about -2.
  • C_I (prefactor in Eq. 6) = ~0.3
    Fit coefficient for power-law decay of mutual information.
  • a_vol (volume-law coefficient, Eq. 7) = varies with h; approaches 1 in ergodic regime
    Coefficient in self-entanglement fit <E_A> = a_vol l + a_crit log2 l + a_area, extracted by least squares.
  • a_crit (log-law coefficient, Eq. 7) = peaks near hc
    Coefficient in self-entanglement fit; its peak near transition is used as corroborating evidence.
  • a_area (area-law coefficient, Eq. 7) = dominates in MBL regime
    Coefficient in self-entanglement fit; reflects area-law behavior in the localized phase.
assumptions (5)
  • domain assumption The random-field Heisenberg chain (Eq. 1) hosts the MBL transition for uniform disorder h around 3.5-5, with the L=20 critical point near 3.25.
    Used to interpret h sweeps and select h=3.25 as critical. This is a standard model, but the precise critical point for L=20 is a numerical input.
  • domain assumption A single eigenvector at the middle of the spectrum of each disorder instance represents infinite-temperature behavior.
    Standard in MBL numerics; the paper averages at least 100 instances but only one eigenstate per instance is evaluated in the Results section.
  • domain assumption The TNSLQ method from Refs. [72,73] computes logarithmic negativity accurately for blocks with 2l>12.
    Used without independent validation in this paper; large-block data points in Fig. 4 depend on this approximation.
  • ad hoc to paper Scale invariance at a critical point implies data collapse of normalized quantities versus d/l.
    The interpretive premise that makes the observed collapse evidence for scale invariance rather than a numerical coincidence.
  • domain assumption Disorder averaging over at least 100 noise realizations is sufficient for convergence of the averaged quantities.
    The paper states 'unless explicitly noted average over many (at least 100) different noise realizations', but does not demonstrate convergence explicitly.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Scale Invariant Entanglement Negativity at the Many-Body Localization Transition." pith.science (2026). https://pith.science/paper/S3I7IPE3

@misc{pith2026190802761,
  author       = {Pith},
  title        = {Pith review of: Scale Invariant Entanglement Negativity at the Many-Body Localization Transition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S3I7IPE3}},
  note         = {Machine review of arXiv:1908.02761}
}
read the original abstract

The exact nature of the many-body localization transition remains an open question. An aspect which has been posited in various studies is the emergence of scale invariance around this point, however the direct observation of this phenomenon is still absent. Here we achieve this by studying the logarithmic negativity and mutual information between disjoint blocks of varying size across the many-body localization transition. The two length scales, block sizes and the distance between them, provide a clear quantitative probe of scale invariance across different length scales. We find that at the transition point, the logarithmic negativity obeys a scale invariant exponential decay with respect to the ratio of block separation to size, whereas the mutual information obeys a polynomial decay. The observed scale invariance of the quantum correlations in a microscopic model opens the direction to probe the fractal structure in critical eigenstates using tensor network techniques and provide constraints on the theory of the many-body localization transition.

Figures

Figures reproduced from arXiv: 1908.02761 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Genuine multipartite entanglement as a probe of many-body localization in disordered spin chains with Dzyaloshinskii-Moriya interactions

    quant-ph 2025-07 conditional novelty 5.0 of 10

    Quenched genuine multipartite entanglement (GGM) tracks the ergodic-MBL transition in disordered Heisenberg chains, with three-body DM interactions delaying localization to higher disorder strengths.

Reference graph

Works this paper leans on

84 extracted references · 79 canonical work pages · cited by 1 Pith paper

  1. [73]

    Fast Computation of Many-Body Entanglement

    J. Gray, “Fast computation of many-body entanglement,” arXiv preprint arXiv:1809.01685 , 2018

    work page Pith review arXiv 2018

  2. [1]

    Sachdev, Quantum phase transitions

    S. Sachdev, Quantum phase transitions . Cambridge Uni- versity Press, 2011

    2011

  3. [2]

    Manipulat- ing quantum entanglement with atoms and photons in a cavity,

    J.-M. Raimond, M. Brune, and S. Haroche, “Manipulat- ing quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. , vol. 73, no. 3, p. 565, 2001

    2001

  4. [3]

    Scaling of entanglement close to a quantum phase transition,

    A. Osterloh, L. Amico, G. Falci, and R. Fazio, “Scaling of entanglement close to a quantum phase transition,” Nature, vol. 416, no. 6881, p. 608, 2002

    2002

  5. [4]

    Entanglement in a simple quantum phase transition,

    T. J. Osborne and M. A. Nielsen, “Entanglement in a simple quantum phase transition,” Phys. Rev. A , vol. 66, no. 3, p. 032110, 2002

    2002

  6. [5]

    Entanglement spectrum, critical exponents, and order parameters in quantum spin chains,

    G. De Chiara, L. Lepori, M. Lewenstein, and A. Sanpera, “Entanglement spectrum, critical exponents, and order parameters in quantum spin chains,” Phys. Rev. Lett. , vol. 109, no. 23, p. 237208, 2012

    work page 2012

  7. [6]

    En- tanglement probe of two-impurity kondo physics in a spin chain,

    A. Bayat, S. Bose, P. Sodano, and H. Johannesson, “En- tanglement probe of two-impurity kondo physics in a spin chain,” Phys. Rev. Lett. , vol. 109, no. 6, p. 066403, 2012

    work page 2012

  8. [7]

    Entan- glement structure of the two-channel kondo model,

    B. Alkurtass, A. Bayat, I. Affleck, S. Bose, H. Johannes- son, P. Sodano, E. S. Sørensen, and K. Le Hur, “Entan- glement structure of the two-channel kondo model,” Phys. Rev. B, vol. 93, no. 8, p. 081106, 2016

    work page 2016

Show all 84 references
  1. [8]

    Scaling of entanglement between separated blocks in spin chains at criticality,

    H. Wichterich, J. Molina-Vilaplana, and S. Bose, “Scaling of entanglement between separated blocks in spin chains at criticality,” Phys. Rev. A , vol. 80, no. 1, p. 010304, 2009

    2009

  2. [9]

    Critical and noncritical long-range entanglement in klein- gordon fields,

    S. Marcovitch, A. Retzker, M. Plenio, and B. Reznik, “Critical and noncritical long-range entanglement in klein- gordon fields,” Phys. Rev. A , vol. 80, no. 1, p. 012325, 2009

    2009

  3. [10]

    Entangle- ment in quantum critical phenomena,

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “Entangle- ment in quantum critical phenomena,” Phys. Rev. Lett. , vol. 90, no. 22, p. 227902, 2003

    2003

  4. [11]

    Entanglement in the xy spin chain,

    A. R. Its, B.-Q. Jin, and V. E. Korepin, “Entanglement in the xy spin chain,” Journal of Physics A: Mathematical and General, vol. 38, no. 13, p. 2975, 2005

    2005

  5. [12]

    Entanglement entropy and quantum field theory,

    P. Calabrese and J. Cardy, “Entanglement entropy and quantum field theory,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2004, no. 06, p. P06002, 2004

    2004

  6. [13]

    Entanglement neg- ativity in extended systems: a field theoretical approach,

    P. Calabrese, J. Cardy, and E. Tonni, “Entanglement neg- ativity in extended systems: a field theoretical approach,” Journal of Statistical Mechanics: Theory and Experiment , vol. 2013, no. 02, p. P02008, 2013

    2013

  7. [14]

    Negativity spectrum in 1d gapped phases of matter,

    G. B. Mbeng, V. Alba, and P. Calabrese, “Negativity spectrum in 1d gapped phases of matter,” Journal of Physics A: Mathematical and Theoretical , vol. 50, no. 19, p. 194001, 2017

    2017

  8. [15]

    Topological entanglement entropy,

    A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys. Rev. Lett. , vol. 96, no. 11, p. 110404, 2006

    2006

  9. [16]

    An order parameter for impurity systems at quantum criti- cality,

    A. Bayat, H. Johannesson, S. Bose, and P. Sodano, “An order parameter for impurity systems at quantum criti- cality,” Nat. Commun. , vol. 5, p. 3784, 2014

    2014

  10. [17]

    Localization-protected quantum order,

    D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, “Localization-protected quantum order,” Phys. Rev. B , vol. 88, p. 014206, Jul 2013

    2013

  11. [18]

    Hilbert-Glass Transition: New Universal- ity of Temperature-Tuned Many-Body Dynamical Quan- tum Criticality,

    D. Pekker, G. Refael, E. Altman, E. Demler, and V. Oganesyan, “Hilbert-Glass Transition: New Universal- ity of Temperature-Tuned Many-Body Dynamical Quan- tum Criticality,” Phys. Rev. X , vol. 4, p. 011052, Mar 2014

    2014

  12. [19]

    Lo- calization and topology protected quantum coherence at the edge of hot matter,

    Y. Bahri, R. Vosk, E. Altman, and A. Vishwanath, “Lo- calization and topology protected quantum coherence at the edge of hot matter,” Nature communications, vol. 6, p. 7341, 2015

    2015

  13. [20]

    Many-body localization and symmetry-protected topological order,

    A. Chandran, V. Khemani, C. R. Laumann, and S. L. Sondhi, “Many-body localization and symmetry-protected topological order,” Phys. Rev. B , vol. 89, p. 144201, Apr 2014

    2014

  14. [21]

    Many- body localization in a disordered quantum Ising chain,

    J. A. Kj¨ all, J. H. Bardarson, and F. Pollmann, “Many- body localization in a disordered quantum Ising chain,” Phys. Rev. Lett. , vol. 113, no. 10, p. 107204, 2014

    2014

  15. [22]

    Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice,

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, “Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice,” Science, vol. 349, no. 6250, pp. 842–845, 2015

    2015

  16. [23]

    Exploring the many-body localization transition in two dimensions,

    J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, “Exploring the many-body localization transition in two dimensions,” Science, vol. 352, no. 6293, pp. 1547–1552, 2016

    2016

  17. [24]

    Signatures of many-body localization in a controlled open quantum system,

    H. P. L¨ uschen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch, and U. Schneider, “Signatures of many-body localization in a controlled open quantum system,” Phys. Rev. X , vol. 7, no. 1, p. 011034, 2017

    2017

  18. [25]

    Observation of many- body localization in a one-dimensional system with a single-particle mobility edge,

    T. Kohlert, S. Scherg, X. Li, H. P. L¨ uschen, S. D. Sarma, I. Bloch, and M. Aidelsburger, “Observation of many- body localization in a one-dimensional system with a single-particle mobility edge,” Phys. Rev. Lett. , vol. 122, no. 17, p. 170403, 2019

    2019

  19. [26]

    Many-body localization in a quantum simulator with pro- grammable random disorder,

    J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, “Many-body localization in a quantum simulator with pro- grammable random disorder,” Nat. Phys. , vol. 12, no. 10, p. 907, 2016

    2016

  20. [27]

    Emulating many-body localization with a superconducting quantum processor,

    K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song, W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, et al. , “Emulating many-body localization with a superconducting quantum processor,” Phys. Rev. Lett. , vol. 120, no. 5, p. 050507, 2018

    2018

  21. [28]

    Propagation and localization of collective excitations on a 24-qubit superconducting processor,

    Y. Ye, Z.-Y. Ge, Y. Wu, S. Wang, M. Gong, Y.-R. Zhang, Q. Zhu, R. Yang, S. Li, F. Liang, et al. , “Propagation and localization of collective excitations on a 24-qubit superconducting processor,” Phys. Rev. Lett. , vol. 123, no. 5, p. 050502, 2019

    2019

  22. [29]

    Metal–insulator transition in a weakly interacting many- electron system with localized single-particle states,

    D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal–insulator transition in a weakly interacting many- electron system with localized single-particle states,” Ann. Phys., vol. 321, pp. 1126–1205, May 2006. 7

    2006

  23. [30]

    Localization of interacting fermions at high temperature,

    V. Oganesyan and D. A. Huse, “Localization of interacting fermions at high temperature,” Phys. Rev. B , vol. 75, no. 15, p. 155111, 2007

    2007

  24. [31]

    Many-body localization phase transition,

    A. Pal and D. A. Huse, “Many-body localization phase transition,” Phys. Rev. B , vol. 82, no. 17, p. 174411, 2010

    2010

  25. [32]

    Many-Body Localiza- tion and Thermalization in Quantum Statistical Mechan- ics,

    R. Nandkishore and D. A. Huse, “Many-Body Localiza- tion and Thermalization in Quantum Statistical Mechan- ics,” Annual Review of Condensed Matter Physics , vol. 6, pp. 15–38, 2015

    2015

  26. [33]

    Col- loquium: Many-body localization, thermalization, and entanglement,

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Col- loquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. , vol. 91, p. 021001, May 2019

    2019

  27. [34]

    Many-body localization,

    A. Pal, “Many-body localization,” 2012

    2012

  28. [35]

    Area laws in a many-body localized state and its implications for topological order,

    B. Bauer and C. Nayak, “Area laws in a many-body localized state and its implications for topological order,” Journal Of Statistical Mechanics-Theory And Experiment , vol. 2013, p. P09005, sep 2013

    2013

  29. [36]

    Un- bounded Growth of Entanglement in Models of Many- Body Localization,

    J. H. Bardarson, F. Pollmann, and J. E. Moore, “Un- bounded Growth of Entanglement in Models of Many- Body Localization,” Phys. Rev. Lett. , vol. 109, p. 017202, July 2012

    2012

  30. [37]

    Universal slow growth of entanglement in interacting strongly disordered systems,

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, “Universal slow growth of entanglement in interacting strongly disordered systems,” Phys. Rev. Lett. , vol. 110, no. 26, p. 260601, 2013

    2013

  31. [38]

    Phe- nomenology of fully many-body-localized systems,

    D. A. Huse, R. Nandkishore, and V. Oganesyan, “Phe- nomenology of fully many-body-localized systems,” Phys. Rev. B, vol. 90, p. 174202, Nov. 2014

    2014

  32. [39]

    Local conserva- tion laws and the structure of the many-body localized states,

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, “Local conserva- tion laws and the structure of the many-body localized states,” Phys.l Rev. Lett. , vol. 111, no. 12, p. 127201, 2013

    2013

  33. [40]

    Entanglement spreading in a many-body localized system,

    A. Nanduri, H. Kim, and D. A. Huse, “Entanglement spreading in a many-body localized system,” Phys. Rev. B, vol. 90, no. 6, p. 064201, 2014

    2014

  34. [41]

    Many-body localization edge in the random-field heisenberg chain,

    D. J. Luitz, N. Laflorencie, and F. Alet, “Many-body localization edge in the random-field heisenberg chain,” Phys. Rev. B , vol. 91, no. 8, p. 081103, 2015

    2015

  35. [42]

    Many-body localiza- tion transition: Schmidt gap, entanglement length, and scaling,

    J. Gray, S. Bose, and A. Bayat, “Many-body localiza- tion transition: Schmidt gap, entanglement length, and scaling,” Phys. Rev. B , vol. 97, no. 20, p. 201105, 2018

    2018

  36. [43]

    Bimodal entangle- ment entropy distribution in the many-body localization transition,

    X. Yu, D. J. Luitz, and B. K. Clark, “Bimodal entangle- ment entropy distribution in the many-body localization transition,” Phys. Rev. B , vol. 94, p. 184202, Nov 2016

    2016

  37. [44]

    Power-law entanglement spectrum in many-body localized phases,

    M. Serbyn, A. A. Michailidis, D. A. Abanin, and Z. Papi´ c, “Power-law entanglement spectrum in many-body localized phases,” Phys. Rev. Lett., vol. 117, no. 16, p. 160601, 2016

    2016

  38. [45]

    Quantum mutual information as a probe for many-body localization,

    G. De Tomasi, S. Bera, J. H. Bardarson, and F. Pollmann, “Quantum mutual information as a probe for many-body localization,” Phys. Rev. Lett. , vol. 118, p. 016804, Jan 2017

    2017

  39. [46]

    Crit- ical properties of the many-body localization transition,

    V. Khemani, S.-P. Lim, D. Sheng, and D. A. Huse, “Crit- ical properties of the many-body localization transition,” Phys. Rev. X , vol. 7, no. 2, p. 021013, 2017

    2017

  40. [48]

    Critical Properties of the Many-Body Localization Tran- sition,

    V. Khemani, S. P. Lim, D. N. Sheng, and D. A. Huse, “Critical Properties of the Many-Body Localization Tran- sition,” arXiv:1607.05756, July 2016

    2016 arXiv

  41. [49]

    Analytically solvable renormalization group for the many-body local- ization transition,

    A. Goremykina, R. Vasseur, and M. Serbyn, “Analytically solvable renormalization group for the many-body local- ization transition,” Phys. Rev. Lett. , vol. 122, p. 040601, Jan 2019

    2019

  42. [50]

    Kosterlitz-thouless scaling at many-body localization phase transitions,

    P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran, M. Serbyn, and R. Vasseur, “Kosterlitz-thouless scaling at many-body localization phase transitions,” Phys. Rev. B, vol. 99, p. 094205, Mar 2019

    2019

  43. [51]

    Behavior of l-bits near the many-body localization transition,

    A. K. Kulshreshtha, A. Pal, T. B. Wahl, and S. H. Si- mon, “Behavior of l-bits near the many-body localization transition,” Phys. Rev. B , vol. 98, p. 184201, Nov 2018

    2018

  44. [52]

    Multiscale entanglement clusters at the many-body localization phase transition,

    L. Herviou, S. Bera, and J. H. Bardarson, “Multiscale entanglement clusters at the many-body localization phase transition,” Phys. Rev. B , vol. 99, p. 134205, Apr 2019

    2019

  45. [53]

    Volume of the set of separable states,

    K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewen- stein, “Volume of the set of separable states,” Phys. Rev. A, vol. 58, no. 2, p. 883, 1998

    1998

  46. [54]

    Partial teleportation of entanglement in a noisy environment,

    J. Lee, M. Kim, Y. Park, and S. Lee, “Partial teleportation of entanglement in a noisy environment,” J. Mod. Opt. , vol. 47, pp. 2151–2164, Oct. 2000

    2000

  47. [55]

    Computable measure of entanglement,

    G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A , vol. 65, p. 032314, Feb. 2002

    2002

  48. [56]

    Logarithmic negativity: a full entangle- ment monotone that is not convex,

    M. B. Plenio, “Logarithmic negativity: a full entangle- ment monotone that is not convex,” Phys. Rev. Lett. , vol. 95, no. 9, p. 090503, 2005

    2005

  49. [57]

    Negativity as the entanglement measure to probe the kondo regime in the spin-chain kondo model,

    A. Bayat, P. Sodano, and S. Bose, “Negativity as the entanglement measure to probe the kondo regime in the spin-chain kondo model,” Physical Review B, vol. 81, no. 6, p. 064429, 2010

    2010

  50. [58]

    Entanglement negativity after a global quantum quench,

    A. Coser, E. Tonni, and P. Calabrese, “Entanglement negativity after a global quantum quench,” Journal of Statistical Mechanics: Theory and Experiment , vol. 2014, no. 12, p. P12017, 2014

    2014

  51. [59]

    Towards the en- tanglement negativity of two disjoint intervals for a one dimensional free fermion,

    A. Coser, E. Tonni, and P. Calabrese, “Towards the en- tanglement negativity of two disjoint intervals for a one dimensional free fermion,” Journal of Statistical Mechan- ics: Theory and Experiment , vol. 2016, no. 3, p. 033116, 2016

    2016

  52. [60]

    Negativity for two blocks in the one-dimensional spin-1 affleck-kennedy- lieb-tasaki model,

    R. A. Santos, V. Korepin, and S. Bose, “Negativity for two blocks in the one-dimensional spin-1 affleck-kennedy- lieb-tasaki model,” Phys. Rev. A , vol. 84, no. 6, p. 062307, 2011

    2011

  53. [61]

    Universality of the negativity in the lipkin-meshkov-glick model,

    H. Wichterich, J. Vidal, and S. Bose, “Universality of the negativity in the lipkin-meshkov-glick model,” Phys. Rev. A, vol. 81, p. 032311, Mar 2010

    2010

  54. [62]

    Scaling of tripartite entanglement at impurity quantum phase transitions,

    A. Bayat, “Scaling of tripartite entanglement at impurity quantum phase transitions,” Phys. Rev. Lett. , vol. 118, no. 3, p. 036102, 2017

    2017

  55. [63]

    Uni- versal Properties of Many-Body Delocalization Transi- tions,

    A. C. Potter, R. Vasseur, and A. Parameswaran, S., “Uni- versal Properties of Many-Body Delocalization Transi- tions,” Phys. Rev. X , vol. 5, p. 031033, Sept. 2015

    2015

  56. [64]

    Theory of the Many-Body Localization Transition in One-Dimensional Systems,

    R. Vosk, D. A. Huse, and E. Altman, “Theory of the Many-Body Localization Transition in One-Dimensional Systems,” Phys. Rev. X , vol. 5, p. 31032, sep 2015

    2015

  57. [65]

    Many- body localization phase transition: A simplified strong- randomness approximate renormalization group,

    L. Zhang, B. Zhao, T. Devakul, and D. A. Huse, “Many- body localization phase transition: A simplified strong- randomness approximate renormalization group,” Phys. Rev. B, vol. 93, p. 224201, Jun 2016

    2016

  58. [66]

    Scaling theory of entanglement at the many-body localization transition,

    P. T. Dumitrescu, R. Vasseur, and A. C. Potter, “Scaling theory of entanglement at the many-body localization transition,” Phys. Rev. Lett. , vol. 119, p. 110604, Sep 2017

    2017

  59. [67]

    Renormalization-group study of the many-body localization transition in one dimension,

    A. Morningstar and D. A. Huse, “Renormalization-group study of the many-body localization transition in one dimension,” Phys. Rev. B , vol. 99, p. 224205, Jun 2019. 8

    2019

  60. [68]

    Entanglement negativity in random spin chains,

    P. Ruggiero, V. Alba, and P. Calabrese, “Entanglement negativity in random spin chains,” Phys. Rev. B , vol. 94, p. 035152, Jul 2016

    2016

  61. [69]

    Many-body localization edge in the random-field Heisenberg chain,

    D. J. Luitz, N. Laflorencie, and F. Alet, “Many-body localization edge in the random-field Heisenberg chain,” Phys. Rev. B , vol. 91, p. 081103, Feb. 2015

    2015

  62. [70]

    Parallel distributed computing using Python,

    L. D. Dalcin, R. R. Paz, P. A. Kler, and A. Cosimo, “Parallel distributed computing using Python,”Adv. Water Resour., vol. 34, pp. 1124–1139, Sept. 2011

    2011

  63. [71]

    SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigen- value Problems,

    V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigen- value Problems,” ACM Trans. Math. Softw. , vol. 31, pp. 351–362, Sept. 2005

    2005

  64. [72]

    quimb: a python library for quantum infor- mation and many-body calculations,

    J. Gray, “ quimb: a python library for quantum infor- mation and many-body calculations,” Journal of Open Source Software, vol. 3, no. 29, p. 819, 2018

    2018

  65. [74]

    Dis- tributed entanglement,

    V. Coffman, J. Kundu, and W. K. Wootters, “Dis- tributed entanglement,” Physical Review A , vol. 61, no. 5, p. 052306, 2000

    2000

  66. [75]

    Strong monogamy of bipar- tite and genuine multipartite entanglement: the gaussian case,

    G. Adesso and F. Illuminati, “Strong monogamy of bipar- tite and genuine multipartite entanglement: the gaussian case,” Phys. Rev. Lett. , vol. 99, no. 15, p. 150501, 2007

    2007

  67. [76]

    Entanglement between two subsystems, the wigner semi- circle and extreme-value statistics,

    U. T. Bhosale, S. Tomsovic, and A. Lakshminarayan, “Entanglement between two subsystems, the wigner semi- circle and extreme-value statistics,” Phys. Rev. A , vol. 85, no. 6, p. 062331, 2012

    2012

  68. [77]

    Class of quantum many-body states that can be efficiently simulated,

    G. Vidal, “Class of quantum many-body states that can be efficiently simulated,” Phys. Rev. Lett., vol. 101, p. 110501, Sep 2008

    2008

  69. [78]

    Algorithms for entanglement renormalization,

    G. Evenbly and G. Vidal, “Algorithms for entanglement renormalization,” Phys. Rev. B , vol. 79, p. 144108, Apr 2009

    2009

  70. [79]

    Finding Matrix Prod- uct State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians,

    X. Yu, D. Pekker, and B. K. Clark, “Finding Matrix Prod- uct State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians,” Phys. Rev. Lett. , vol. 118, p. 017201, Jan 2017

    2017

  71. [80]

    Obtain- ing Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach,

    V. Khemani, F. Pollmann, and S. L. Sondhi, “Obtain- ing Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach,” Phys. Rev. Lett. , vol. 116, p. 247204, Jun 2016

    2016

  72. [81]

    Spectral tensor networks for many-body localization,

    A. Chandran, J. Carrasquilla, I. H. Kim, D. A. Abanin, and G. Vidal, “Spectral tensor networks for many-body localization,” Phys. Rev. B , vol. 92, p. 024201, Jul 2015

    2015

  73. [82]

    Efficient variational diagonalization of fully many-body localized Hamiltonians,

    F. Pollmann, V. Khemani, J. I. Cirac, and S. L. Sondhi, “Efficient variational diagonalization of fully many-body localized Hamiltonians,” Phys. Rev. B , vol. 94, p. 041116, Jul 2016

    2016

  74. [83]

    Encoding the structure of many-body localization with matrix product operators,

    D. Pekker and B. K. Clark, “Encoding the structure of many-body localization with matrix product operators,” Phys. Rev. B , vol. 95, p. 035116, Jan 2017

    2017

  75. [84]

    Efficient represen- tation of fully many-body localized systems using tensor networks,

    T. B. Wahl, A. Pal, and S. H. Simon, “Efficient represen- tation of fully many-body localized systems using tensor networks,” Phys. Rev. X , vol. 7, p. 021018, May 2017

    2017

  76. [85]

    Signatures of the many-body localized regime in two dimensions,

    T. B. Wahl, A. Pal, and S. H. Simon, “Signatures of the many-body localized regime in two dimensions,” Nature Physics, vol. 15, no. 2, p. 164, 2019

    2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.